Question

If $ \cos \theta = \frac{5}{13} $ and $ \theta $ is in the fourth quadrant, find the value of $ \tan \theta $.

• A. \( -\frac{12}{5} \)
• B. \( \frac{12}{5} \)
• C. \( -\frac{5}{12} \)
• D. \( \frac{5}{12} \)

Hint:

In trigonometric problems, use the quadrant to determine the sign of trigonometric functions and apply identities like \( \sin^2 \theta + \cos^2 \theta = 1 \) to find other values.

Answer 1

The Correct Option is A

Given \( \cos \theta = \frac{5}{13} \) in the fourth quadrant, where \( \cos \theta \) is positive and \( \tan \theta \) is negative. Using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] \[ \sin^2 \theta = 1 - \left( \frac{5}{13} \right)^2 = 1 - \frac{25}{169} = \frac{144}{169} \] \[ \sin \theta = \pm \frac{12}{13} \] Since \( \theta \) is in the fourth quadrant, \( \sin \theta \) is negative: \[ \sin \theta = -\frac{12}{13} \] Now, calculate \( \tan \theta \): \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{12}{13}}{\frac{5}{13}} = -\frac{12}{5} \] Thus, the value of \( \tan \theta \) is: \[ \boxed{-\frac{12}{5}} \]